A FURTHER STUDY OF MULTIVARIATE LIOUVILLE DISTRIBUTION AND SOME RELATED DISTRIBUTIONS

Suppose （X₁, …, X_n） has a multivariate Liouville distribution （MLD） L_na₁, …, α_n; f（·） and Γ（θ,α） is a Gamma distribution. Then Theorem 2.1 For any given r, the distribution of （X₁, …, X_n） is uniquely determined by the distribution of sum from 1 to r （X_i）. Specially, sum from 1 to r（X_i）～Γ（θ, sum from 1 to r （α_i））implies: X₁, …, X_n are independent and X_i～Γ（θ, α_i）i=1, …, n.
Theorem 4.1 If T（·） is a scale invariant statistic. Then the distribution of T（X₁, …, X_n） is independent of f（·）. Also T（X₁, …, X_n） and sum from 1 to r （X_i） are mutually independent.
Thispaper also removes all the explicit and implicit restrictions on characterization of MLD by independence in 1, 2, and gives some other characterization results. Many similar results are obtained about the so called generalized MLD and some related distributions.